Editing Hawking radiation (section)

=== Trans-Planckian problem ===

The [[trans-Planckian problem]] is the issue that Hawking's original calculation includes [[quantum]] particles where the [[wavelength]] becomes shorter than the [[Planck length]] near the black hole's horizon. This is due to the peculiar behavior there, where time stops as measured from far away. A particle emitted from a black hole with a [[Wikt:finite|finite]] [[frequency]], if traced back to the horizon, must have had an [[Infinity|infinite]] frequency, and therefore a trans-Planckian wavelength.

The [[Unruh effect]] and the Hawking effect both talk about field modes in the superficially stationary [[spacetime]] that change frequency relative to other coordinates that are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration that constantly [[Doppler shift]]s the modes.{{Citation needed|date=September 2020}}

An outgoing [[photon]] of Hawking radiation, if the mode is traced back in time, has a frequency that diverges from that which it has at great distance, as it gets closer to the horizon, which requires the wavelength of the photon to "scrunch up" infinitely at the horizon of the black hole. In a maximally extended external [[Schwarzschild metric|Schwarzschild solution]], that photon's frequency stays regular only if the mode is extended back into the past region where no observer can go, so Hawking used a different black hole solution without a past region, one that forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified: a microscopic point right at the moment that the black hole first formed.{{cn|date=February 2025}}

The quantum fluctuations at that tiny point, in Hawking's original calculation, contain all the outgoing radiation. The modes that eventually contain the outgoing radiation at long times are redshifted by such a huge amount by their long sojourn next to the event horizon that they start off as modes with a wavelength much shorter than the Planck length. Since the laws of physics at such short distances are unknown, some find Hawking's original calculation unconvincing.<ref>{{Cite journal | doi = 10.1088/0034-4885/66/6/202|arxiv=gr-qc/0304042| title = Do black holes radiate?| journal =[[Reports on Progress in Physics]]| volume = 66| issue = 6| pages = 943–1008| year = 2003| last1 = Helfer | first1 = Adam D. |bibcode=2003RPPh...66..943H|s2cid=16668175}}</ref><ref>{{Cite journal |last1='t Hooft |first1=Gerard |author-link1=Gerard 't Hooft |doi=10.1016/0550-3213(85)90418-3 |title=On the quantum structure of a black hole |journal=[[Nuclear Physics B]] |volume=256 |pages=727–745 |year=1985 |bibcode=1985NuPhB.256..727T }}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevD.44.1731| pmid = 10014053| title = Black-hole evaporation and ultrashort distances| journal =[[Physical Review D]]| volume = 44| issue = 6| pages = 1731–1739| year = 1991| last1 = Jacobson | first1 = Theodore | bibcode = 1991PhRvD..44.1731J}}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevD.52.4559|pmid=10019680|arxiv=hep-th/9506121| title = Hawking radiation without trans-Planckian frequencies| journal =[[Physical Review D]]| volume = 52| issue = 8| pages = 4559–4568| year = 1995| last1 = Brout | first1 = Robert | last2 = Massar | first2 = Serge | last3 = Parentani | first3 = Renaud | last4 = Spindel | first4 = Philippe |bibcode=1995PhRvD..52.4559B|s2cid=26432764}}</ref>

The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations. The same effect occurs for regular matter falling onto a [[white hole]] solution. Matter that falls on the white hole accumulates on it, but has no future region into which it can go. Tracing the future of this matter, it is compressed onto the final singular endpoint of the white hole evolution, into a trans-Planckian region. The reason for these types of divergences is that modes that end at the horizon from the point of view of outside coordinates are singular in frequency there. The only way to determine what happens classically is to extend in some other coordinates that cross the horizon.

There exist alternative physical pictures that give the Hawking radiation in which the trans-Planckian problem is addressed.{{citation needed|date=August 2016}} The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time.<ref name="witt-blah">For an alternative derivation and more detailed discussion of Hawking radiation as a form of Unruh radiation, see: {{cite book|last=de Witt |first=Bryce |date=1980 |contribution=Quantum gravity: the new synthesis |page=[https://archive.org/details/generalrelativit00isra/page/n711 696] |title=General Relativity: An Einstein Centenary Survey |publisher=Cambridge University Press |url=https://archive.org/details/generalrelativit00isra |url-access=limited |editor1-first=Stephen W. |editor1-last=Hawking |editor2-first=Werner |editor2-last=Israel |isbn=0-521-29928-4}}</ref> In the Unruh effect, the magnitude of the temperature can be calculated from ordinary [[Hermann Minkowski|Minkowski]] field theory, and is not controversial.